Simplify the following expression and state the condition under which the simplification is valid. $p = \dfrac{r^2 - 49}{r - 7}$
First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = r$ $ b = \sqrt{49} = -7$ So we can rewrite the expression as: $p = \dfrac{({r} {-7})({r} + {7})} {r - 7} $ We can divide the numerator and denominator by $(r - 7)$ on condition that $r \neq 7$ Therefore $p = r + 7; r \neq 7$